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ICCAS2003 October 22-25, Gyeongju TEMF Hotel, Gyeongju, Korea

1. INTRODUCTION

A hybrid robot denotes a robot system that isconstructed by combination of serial and parallelmodules or a series of parallel modules. Freeman andTesar[1], Kang, at al[2], and Cho[3] presented dynamicmodeling methodologies for such robotic systmes. Also,Sklar[4] presented a kinematic modeling method forhybrid robotic systems.In general, the Newton-Euler formulation, theLagrangian method, and the open-chain method havebeen employed for dynamic modeling of hybrid roboticsystems. Though these methods are well knownformulation, however, these methods have shortcomingin that it must calculate dynamics of whole manipulatoragain even if the structure of robot changes very a littleand that requires expensive computation as the size ofthe system increases.

Fig. 1 A hybrid robot composed of well knownstructures

For example, even though the dynamics of twoindependent systems, as shown in Fig. 1, are known, thewhole dynamics needs to be reconstructed when the twomodules are combined as one system.In this paper, therefore, we propose an efficientdynamic modeling methodology for hybrid roboticsystems that are constructed by adding another robot tothe existing robot mechanism successively. Thisapproach has advantage in that it is not necessary tocalculate the whole dynamics again.An Efficient Dynamic Modeling Method for Hybrid Robotic Systems

Abstract:In this paper, we deal with the kinematic and dynamic modeling of hybrid robotic systems that are constructedby combination of parallel and serial modules or series of parallel modules. Previously, open-tree structure has beenemployed for dynamic modeling of hybrid robotic systems. Though this method is generally used, however, it requiresexpensive computation as the size of the system increases. Therefore, we propose an efficient dynamic modelingmethodology for hybrid robotic systems. Initially, the dynamic model for the proximal module is obtained with respectto the independent joint coordinates. Then, in order to represent the operational dynamics of the proximal module, wemodel virtual joints attached at the top platform of the proximal module. The dynamic motion of the next module exertsdynamic forces to the virtual joints, which in fact is equivalent to the reaction forces exerted on the platform of thelower module by the dynamics of the upper module. Then, the dynamic forces at the virtual joints are distributed to theindependent joints of the proximal module. For multiple modules, this scheme can be constructed as a recursivedynamic formulation, which results in reduction of the complexness of the open-tree structure method for modeling of

The proposed method is explained as follows. First,the dynamic model for the proximal module is obtainedwith respect to the independent joint coordinates. Then,we represent the operational dynamics of a lowermodule as that of equivalent virtual joints attached atthe top platform of the lower module. Next, the dynamicmodel of the upper module including the virtual jointsand links is obtained. Eventually, the effective dynamicmodel of the lower model is represented as the sum ofvirtual joints’ dynamics and its own dynamic model.Here, the physical meaning of the virtual joints isequivalent to the reaction forces exerted on the platformof the lower module by the dynamic motion of the uppermodule.The basic idea of the proposed method is described inFig. 2.

Fig. 2 The concept of the proposed dynamicmodeling method for hybrid robotic systems

Since two serial-chains have the same velocities atthe center of platform, we have the constraint equationas follows

1 1 2 2[ ] [ ]ss u s s u su G Gφφφφ= =, (4)where

[]sTv vvu x y= Φ  , (5)and denotes the Jacobian and the left subscript ofGdenotes the number of serial sub-chain of this parallelsystem. The superscript and subscript to the right side ofGdenote the dependent and independent parameters,respectively.[ ]GIf we select the independent and dependentparameters as

3. DYNAMICSand the acceleration relation between the output and theinput vector is obtained by differentiating Eq. (30)with respect to time as

[ ] [ ]v v u v v T v u vu G Hφφφφ φφ= +  . (31)3. 1 Dynamics of the hybrid robot by the open-treestructure methodIn this section, we deal with the dynamic formulationof the hybrid robot by the open-tree structure methodthat is well known method for general robot dynamics.By cutting a link as shown in Fig. 5, we have two openchain structures.Finally, by substituting Eq. (15) and Eq. (26) intoEq. (30) and Eq. (31), we can obtain the velocity andthe acceleration of the hybrid robot described in termsof independent joints.Using the Lagrange’s form of d’Alembert principleproposed by Freeman and Tesar[1], the dynamic modelfor the ithchain is represented as

2.3 Kinematics of the hybrid robotThe mobility of the hybrid robot given in Fig. 5 is 6.Thus, six independent inputs are required to drive thesystem. The system can be visualized as two serialchains by cutting a joint or a link.*[ ] [ ]h h h h T h hi i i i iI Pφ φφ φφφ*iτ φφφ= +  , (39)wherehiφτrepresents the torque vector applied to the

ICCAS2003 October 22-25, Gyeongju TEMF Hotel, Gyeongju, Korea

joints of the ithserial sub-chain.*[i]Iφφand*[ ]iPφφφ

are the joint-referenced effective inertia matrix and theeffective inertia power array relating to centrifugal andCoriolis forces, respectively.The dynamic model for the six independent joints isobtained by virtual work principle between theLagrangian coordinates and the minimum independentjoints as[5]

3. 2 Dynamics of the 6-bar linkageThe dynamic model of the 6-bar linkage, shown inFig. 3, can be also obtained using the open-tree structuremodel formed by cutting the center of the platform.The dynamic model for the three independent jointsis obtained by virtual work principle between theLagrangian coordinates and the minimum independentjoints as[5]

[ ]ss a T su uGaττ=(46)the dynamic model in the operational space can be alsoobtained. In Eq. (46),suτis the operational torquevector and[ ]sauGis the inverse matrix of the forwardJacobian[ ]suaG.

3.3 Efficient dynamic model of a hybrid robotIn the proposed method, it is assumed that the wholekinematic and dynamic models both the six-bar linkageand the 3DOF serial module are already given.Now, the dynamic model of the upper moduleincluding the virtual joints will be described. Thedynamic equation of the upper module is described as

[ ]s u T v sh a v aavaGττττ⎡⎤+=⎢⎥⎣⎦. (48)It implies that the dynamic equation of a hybrid robotcan be easily derived by union of the dynamic model ofthe augmented virtual joints and those of the two givenmodules.The proposed scheme is useful in that it is able to usethe dynamic models of the given modules consisting ofthe hybrid robot. To examine its meaning, Eq. (47) isrewritten as

saha vaφφφ⎡ ⎤=⎢ ⎥⎢ ⎥⎣ ⎦, (52)andhCis the vector regarding nonlinear velocities.It is observed from Eq. (51) that the moduledynamics given by*[ ]saaIand are still used inthis dynamic model, and that only the termsand denoting the dynamic model of the virtualjoints need to be calculated additionally.*[ ]aaIφφ*[ ]vvIφφ*[ ]vaIφφConclusively, instead of calculating the dynamics ofthe open tree structure, given by Eq. (40), the proposedmethod is found to be computationally efficient.Furthermore, it is very appealing in that the dynamicmodels of the given modules are still used instead ofdeveloping totally new dynamic model.ICCAS2003 October 22-25, Gyeongju TEMF Hotel, Gyeongju, Korea

4. SIMULATION

In this section, simulation for inverse dynamics isperformed to validate the proposed modeling algorithm.The goal of this section is to show that torque values ofthe independent joints of the hybrid robot obtained bythe proposed method are the same as those of thedynamic model employing the open-tree structuremethod.Fig. 7 Torque values obtained by the proposed method

A given operational trajectory shown in the Fig. 6 isa circle whose the radius( ) is 5cm. The center point(r,c cxy) and an initial point of the end-effecter areobtained by initial joint angles given in Eq. (53) and(54). The rotational angle( ) between the distal linkand the x-axis is fixed as . And the trajectory isfollowed by the robot in four seconds. The initial andlast velocities of the end-effecter are zero. The abovementioned trajectory can be represented as followsΦ90

1[120 -60 -60 60 60 -30 ]hφ=     T, (53)For multiple modules, the proposed scheme can beexpanded as a recursive dynamic formulation, whichresults in reduction of the complexness of the open-treestructure method for the modeling of hybrid roboticsystems. As shown in Fig. 9, the first-order kinematicsand dynamics of the hybrid robotic systems that consistof ‘n’ modules are obtained by adding upper module’sdynamics to the base module’s dynamics one by one.

When all lengths and masses of links are given 0.4and 0.1, respectively, torque values obtained by theproposed method and the open-tree structure method arerepresented in Fig. 7 and Fig. 8, respectively.mkgAs shown in Fig. 7 and 8, torque values obtained bythe two methods are identical. The computational effortof the proposed method is much less than that of theopen-chain method since it is not necessary to calculatethe whole dynamics again, which was the case of theusual open-chain dynamic modeling methodology.Namely, it validates that the proposed method is anefficient dynamic modeling method for hybrid roboticsystems.Fig. 9 An equivalent dynamic model of a hybridrobot constructed by ‘n’ modules

5. 1 Kinematics

The first-order kinematics of the (i-1)thmodule canbe described, from Eq. (36), as

where1iu−,1hi aφ−, and are the output velocityvector of the i-1thmodule, the total independent jointvelocity vector of the whole module up to i-1thmodule,and the Jacobian that relates the output vector to thetotal independent joints up to i-1thmodule. Also, theoutput velocity vector of the ithmodule can be expressedin terms of the ithvirtual joints and the independentjoints of the ithmodule as1[h ui aG−

1[ ]h u T hh i a i v i ai ai aG1ττττ−−⎡⎤+=⎢⎥⎣⎦, (67)]

[ ]i vui i vai au Gφφ⎡⎤=⎢⎥⎢⎥⎣⎦, (60)wherei vτdenotes the dynamics at the virtual joints tosupport the dynamic motion of the ithmodule,i aτ

denotes the dynamics corresponding to the independentjoints of the ithmodule itself, and1hi aτ−denotes thedynamics of the (i-1) modules, respectively.

6. CONCLUSIONS

wherei vφ,i aφ, and are the velocity vectors ofthe ithvirtual joints and the independent joints of the ith

module, and the Jacobian of the ithmodule includingvirtual joints, respectively. Since the virtual joints’velocity of the ithmodule is identical to the outputvelocity of the (i-1)thmodule, Eq. (60) can be writtenas[ ]ui vaGIn this paper, we proposed a new efficient dynamicmodeling methodology for hybrid robotic system andvalidated the proposed method by comparing torquevalues of the proposed method with those of the existingmethod. Our future work is to apply the proposedmethod to general type of hybrid robotic mechanismsand design modular type hybrid robotic systems.

W. Cho, "Development of a dynamic modelingtechnique and its application to the analysis andcontrol of a high precision robotic manipulator,”Ph.D. Dissertation, Department of MechanicalEngineering, The University of Texas at Austin, TX,1989.[4]

The dynamic equationhi aτfor a hybrid roboticsystem that is composed of ‘i’ modules is obtained byreiteration of the process proposed in section 3.3.The dynamics of the first module is given by

1 1haaττ=, (65)[5]

B.-J. Yi, “Analysis of redundantly actuatedmechanisms with applications to design and controlof advanced robotic systems,” Ph.D. Dissertation,Department of Mechanical Engineering, TheUniversity of Texas at Austin, TX, 1991.where1 aτdenotes the motion torque with respect tothe independent joints of the first module itself.The dynamics of the second module is given by

1 2 122[ ]h u T hha v aaaGττττ⎡ ⎤+=⎢⎣ ⎦⎥, (66)where2 vτand2 aτdenotes the motion torques at thevirtual joints and the independent joints of the secondmodule itself, respectively.Conclusively, the dynamicshi aτof the hybridrobotic system that consists of ‘i’ modules is obtained as